Hi, I’m using the comment after Theorem 10.9 on page 396 of Apostol. The comment tells us that if

then we can conclude that the convergence of implies the convergence of . The idea of this (without giving a formal proof) is just that the must be smaller than the for all (otherwise the limit couldn’t be going to 0). As you point out, the full theorem doesn’t hold in this case since might converge, but this would not imply the convergence of .

In this case though we are saying that if converged it would imply that converges. Since we know does not converge, this means cannot converge either. Does that make sense?

Ha, thanks, and no problem. I think you can get to a formal proof of the comment by following Apostol’s proof of the theorem, but leaving off one side of the inequality. So, since we know there is some such that for all we have (just taking in the definition of the limit) and so for all . So, then by Theorem 10.8 we have that converges implies converges. (This still isn’t totally rigorous, but it’s closer.)